A Crank-Nicolson type space-time finite element method for computing on moving meshes
A variant of the continuous finite element method suitable for computation on deforming meshes is introduced. The method is an extension of the Crank-Nicolson method, and computational experience confirms that the method retains the second-order temporal accuracy of the Crank-Nicolson method. The exact conditions necessary for second order accuracy with regards to the deformation of the mesh remains to be investigated. The interest in space-time finite element methods for flows in deforming domains is motivated by the fact that it allows for mimicking the use of Lagrangian coordinates without introducing actual differencing along the characteristics. This means that if the velocity field used to convect the mesh is different from the actual velocity field (which is the usual situation), there will remain a small residual convective term. The resulting nonsymmetry of the corresponding matrix problem is the price paid for consistency; the suggested method will work also if it is difficult to align the mesh along the characteristics. Furthermore, by coupling the approximation of the solution to the approximation of the flow field, there is an immediate gain when the method is applied to nonlinear flow problems where the flow field is identical to the solution, as, e.g., in the Navier-Stokes equations. Clearly, in such a case the residual convective term becomes identical to zero by definition, which automatically reduces (the discretized) Navier-Stokes equation to the Stokes equation without further approximations.
scientific article; zbMATH DE number 1443372
scientific article